/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 62 Evaluate the following limits or... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 62

Evaluate the following limits or explain why they do not exist. Check your results by graphing. $$\lim _{x \rightarrow 0}\left(e^{5 x}+x\right)^{1 / x}$$

Short Answer

Expert verified
Answer: The limit of the given expression as x approaches 0 is \(e^6\).

Step by step solution

01

Rewrite the limit expression with property of exponentiation

We start by rewriting the limit using the property of exponentiation: $$\lim _{x\rightarrow 0} e^{(\ln((e^{5x} + x)^{1/x}))} = e^{\lim _{x \rightarrow 0}\ln((e^{5x} + x)^{1 / x})}$$ This allows us to focus on finding the limit of the exponent: $$\lim _{x \rightarrow 0}\ln((e^{5x} + x)^{1 / x})$$
02

Apply L'H么pital's Rule for indeterminate forms

The expression is still an indeterminate form, so we'll need to apply L'H么pital's Rule. To do that, we should rewrite the limit expression like this: $$\lim _{x \rightarrow 0}\frac{\ln(e^{5x} + x)}{x}$$ Now we can apply L'H么pital's Rule: $$\lim _{x \rightarrow 0}\frac{\frac{d}{\text d x} \ln(e^{5x} + x)}{\frac{d}{\text d x} x}$$
03

Differentiate the numerator and denominator

Differentiating both numerator and denominator, we get: $$\frac{d}{\text d x} \ln(e^{5x} + x) = \frac{(5e^{5x} + 1)}{(e^{5x} + x)}$$ $$\frac{d}{\text d x} x = 1$$ So the limit expression now becomes: $$\lim_{x\rightarrow 0} \frac{(5e^{5x} + 1)}{(e^{5x} + x)}$$
04

Plug in \(x = 0\) to find the limit

Now, we can plug in \(x = 0\) into the expression (since it's no longer indeterminate): $$\lim_{x\rightarrow 0} \frac{(5e^{5(0)} + 1)}{(e^{5(0)} + 0)} = \frac{5+1}{1} = 6$$
05

Plug the limit back into the original expression

Now that we have found the limit of the exponent, we can plug it back into the original expression to find the limit of the whole function: $$\lim _{x \rightarrow 0}\left(e^{5 x}+x\right)^{1 / x} = e^{\lim _{x \rightarrow 0}\ln((e^{5x} + x)^{1 / x})} = e^6$$ So the limit of the given expression as \(x \rightarrow 0\) is \(e^6\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=-32 ; v(0)=20, s(0)=0$$

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=4 ; v(0)=-3, s(0)=2$$

Differentials Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\) $$f(x)=2-a \cos x, a \text { constant }$$

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow \infty} \frac{\log _{2} x}{\log _{3} x}$$

Evaluate the following limits in terms of the parameters a and b, which are positive real numbers. In each case, graph the function for specific values of the parameters to check your results. $$\lim _{x \rightarrow 0}(1+a x)^{b / x}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.